High speed circuit designs, such as signal and/or power integrity designs, package designs, and printed circuit board designs, rely on computer implemented simulation tools. The simulation tools include various modeling algorithms, which should be accurate and efficient over a broad frequency range in order to be useful. Transmission line modeling, in particular, is critical in circuit simulations, including how to accurately and efficiently model conductor surface roughness effect over a broad range of frequencies.
Generally, there are two categories of conventional methods for modeling the conductor surface roughness effect: numerical models and analytical (or empirical) models. Numerical models calculate the conductor surface roughness effect through numerical electromagnetic simulations on specific roughness structures, but are typically too computationally expensive to be incorporated into circuit simulations. Analytical models, which are more appropriate for circuit simulations, calculate a correction factor that is applied to the surface impedance of a smooth surface to take into account power loss and signal delay due to conductor surface roughness. Analytical models for modeling surface roughness include the Hammerstad model, e.g., described by Hammerstad et al., “Accurate Models for Microstrip Computer Aided Design,” IEEE MTT-S INT. MICROW. SYMP. DIG., pp. 407-409 (May 1980); the hemispherical model, e.g., described Hall et al., “Multigigahertz Causal Transmission Line Modeling Methodology Using a 3-D Hemispherical Surface Roughness Approach,” IEEE TRANS. MTT, Vol. 55, No. 12, pp. 2614-2624 (December 2009), and Hall et al., “Advanced Signal Integrity for High-Speed Digital Designs,” JOHN WILEY & SONS, INC. (2009); and the Huray model, e.g., described by Huray, “The Foundations of Signal Integrity,” WILEY-IEEE PRESS (November 2009), each of which is hereby incorporated by reference.
The Hammerstad model loses accuracy at high frequencies. For example, for the roughest copper commonly used in the industry, the Hammerstad starts to break down after about 5 GHz. The hemispherical model works for a broader range of frequencies than the Hammerstad model, but still loses accuracy as frequency increases and cannot fulfill the need for high-speed circuit design. The Huray model may provide relatively good accuracy for frequencies up to 50 GHz. However, the accuracy is dependant on certain model parameter values, such as numbers and sizes of metallic balls (e.g., referred to as “snowballs”), and these values are not easily determined. The appropriate combination of these model parameter values providing the best accuracy must be determined empirically, by trying various combinations and comparing the corresponding simulation results against measured data (e.g., insertion loss). The optimal combination will also vary from one particular surface roughness profile to another. Accordingly, none of the conventional models provides efficient and reliable modeling, particularly with respect to rough conductor surfaces, for a broad range of frequencies.